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## Flow, circulation and flux.

The curl of a plane vector field has been already defined in section  5.

If the vector field represent the velocity field of a fluid, the integral in  8.11 defines the flow of the vector field along the curve .

Definition 8.14   Let be a smooth curve given by the parametrization , where . We denote by a continuous velocity field. The flow of along is equal to

This integral is called a flow integral.

If is closed loop, i.e. , then the flow is called the circulation of around .

Remark 8.15   This definition is valid for a vector field and a curve in the plane.

Example 8.16 (In the plane.)   Take and let be the curve with parametrization , where .

• Evaluate on :

• Tangent:

• Scalar product:

• Integrate:

Example 8.17 (In the space.)   Take and let be the curve with parametrization , where (a loop of an helix, as displayed on Fig.  1).
• Evaluate on :

• Tangent:

• Scalar product:

• Integrate:

Definition 8.18   Let be a smooth curve in the plane. At every point of is defined a unit normal vector pointing outwards. Let now be a vector field in the plane defined over a domain containing the curve .

The flux of across the curve is the line integral

Example 8.19   Take and let be the curve with parametrization , where .

On the curve :

• and ;
• ;
• .
Therefore:

And we have:

Next: Green's theorem in the Up: Vector Fields, Work, Circulation Previous: Work of a force.
Noah Dana-Picard
2001-05-30